Operator entanglement in $\mathrm{SU}(2)$-symmetric dissipative quantum many-body dynamics

1. Same as in first version

1. The improvements in the resubmitted version are very minor, and the main weakness remains (lack of analytical insights)

Points raised in my first report that I feel have been answered unsatisfactorily:

1. In his answer to my comment, the author writes

'We would like to note that the definition of symmetry-resolved operator entanglement in Ref. [59] is actually the same as the one introduced in our manuscript and Ref. [38].'

That is not correct, because the action of the charge operator considered in [38] and [59] is not the same: for a charge operator $Q = Q_A + Q_B$, Ref. [38] considers the case of a density matrix that satisfies

Also, the way the author refers to previous works on this topic in the new version is still very confusing. The author writes:

'It is useful to also introduce the symmetry-resolved operator entanglement, which has attracted extensive attention recently [56-61].'

But Refs. [56,57,58] considered state entanglement, not operator entanglement. Refs. [59,61] did consider operator entanglement, but the definition of the symmetry resolution is different, as explained above. Of all the references cited by the author here, only Ref. [60] is actually about the right topic.

1. In his report the author insists that the situation is more complicated than in the U(1) case of Ref. [38]. That is fine. But then, about the limit of strong dissipation, he writes:

'Moreover, since our model contains dissipation proportional to the dipole interaction between neighbor sites, even in the strong dissipation limit it is still hard to identify the density matrix ρ satisfying $\mathcal{L}_0 (\rho) = \sum_i (L_i \rho L_i^\dagger - { L_i^\dagger L_i , \rho }/2) = 0$ and the corresponding projector $P$ onto these states, which makes the usual perturbation theory also unrealistic for our model.'

I do not understand this comment. I would think that, since the dissipators $L_i$ are the projectors on the singlet on sites $i$ and $i+1$, they generate the whole commutant of SU(2) in the representation $(1/2)^{\otimes N}$, so by standard uniqueness theorem [A. Frigerio, 'Quantum dynamical semigroups and approach to equilibrium', Lett. Math. Phys. 2, 79 (1977)], [Evans, 'Irreducible quantum dynamical semigroups', Commun. Math. Phys. 54 293 (1977)], [Spohn, ' An algebraic condition for the approach to equilibrium of an open N-level system', Letters in Mathematical Physics, 2, 33-38 (1977)] the steady state must be unique and it must be the projector on the spin-$0$ representation of SU(2). Is it not the case? If not, why?

Ask for minor revision

Accept in alternative Journal (see Report)

1- The author has addressed the required changes. In particular the addition in terms of a convergence discussion gives the numerical results now sufficient credibility.

1- All 3 reports have highlighted the lack of a clearer analytical/intuitive understanding.

1- I still don't fully agree with the terminology "measure for classical simulability", even if it is meant exclusively for MPDO representations. "Measure" suggests that a direct connection is mathematically established. In the case of the Shannon entropy used in Eq. (3) this is not entirely clear. See e.g. discussion in the original paper concerning MPS Simulability [Schuch et al., PRL 100, 030504 (2008)].

2- Minor typos: "later" -> "latter" (Appendix)

Accept in alternative Journal (see Report)